# पृष्ठम्:गणितसारसङ्ग्रहः॒रङ्गाचार्येणानूदितः॒१९१२.djvu/३६७

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171
CHAPTER VI--MIXED PROBLEMS.

product is divided by six. (To this resulting quotient), the square of the first term and the (continued product) of the number of terms as diminished by one, the first term, and the common difference, are added. The whole (of this) multiplied by the number of terms becomes the required result.

Examples in illustration thereof.

300. In a series in arithmetical progression), the first term is 3, the common difference is 5. the number of terms is 5. Give out the sum of the squares (of the terms) in the series. (Similarly, in another series), 5 is the first term, B the common difference, and 7 the number of terms. What is the sum of the squares (of the terms) in this series ?

The rule for arriving at the sum of the cubes (of a given number of natural numbers):-

301.[1] The quantity represented by the square of half the (given) number of terms is multiplied by the square of the sum of one and the number of terms. In this (science of) arithmetic, this result is said to be the sum of the cubes (of the given number of natural numbers) by those who know the secret of calculation.

Examples in illustration thereof.

302. Give out (in each case) the sum of the cubes of (the natural numbers up to) 6, 8, 7, 25 and 256.

The rule for arriving at the sum of the cubes (of the terms in a series in arithmetical progression), the first term, the common difference, and the number of terms whereof are optionally chosen:-

303.[2] The sum (of the simple terms in the given series), as multiplied by the first term (therein), is (further) multiplied by the

301.^  Algebraically ${\displaystyle \left({\tfrac {n}{2}}\right)^{2}(n+1)^{2}=s_{3}}$, which sum the cubes of the natural numbers up to n.

303.^  Algebraically,${\displaystyle \pm sa(a\sim b)+s^{2}b=}$ the sum of the cubes of the terms in a series in arithmetical progression, where ${\displaystyle s=}$ the sum of the simple terms of the series. The sign of the first term in the formula is ${\displaystyle +{\text{ or }}-}$ according as ${\displaystyle a>{\text{ or }}